Today, computer aided engineering (CAE) is used for supporting engineers in tasks such as analysis, simulation, design, manufacture, etc. In a conventional engineering design procedure, CAE analysis (e.g., finite element analysis (FEA), finite difference analysis, meshless analysis, computational fluid dynamics (CFD) analysis, modal analysis for reducing noise-vibration-harshness (NVH), etc.) has been employed to evaluate responses (e.g., stresses, displacements, etc.). Using automobile design as an example, a particular version or design of a car is analyzed using FEA to obtain the responses due to certain loading conditions. Engineers will then try to improve the car design by modifying certain parameters or design variables (e.g., thickness of the steel shell, locations of the frames, etc.) based on specific objectives and constraints. Another FEA is conducted to reflect these changes until a “best” design has been achieved. However, this approach generally depends on knowledge of the engineers or based on a trial-and-error method.
Furthermore, as often in any engineering problems or projects, these objectives and constraints are generally in conflict and interact with one another and design variables in nonlinear manners. Thus, it is not very clear how to modify them to achieve the “best” design or trade-off. This situation becomes even more complex in a multi-discipline optimization that requires several different CAE analyses (e.g., FEA, CFD and NVH) to meet a set of conflicting objectives. To solve this problem, a systematic approach to identify the “best” design, referred to as engineering design optimization, is used.
The optimization of such systems with more than one design objective functions is referred to as multi-objective engineering design optimization, which results in a set of optimal engineering designs that represent different trade-offs among design objectives. These optimal engineering designs are referred to as Pareto Optimal Points (POPs) in an N-dimensional design variable space, where N is the number of design variables of interest in the optimization.
One typical prior art approach of this engineering optimization procedure includes the following steps:                a. Select a suitable set of sampling points (i.e., alternative engineering designs) in the N-dimensional design variable space.        b. Conduct numerical simulations (e.g., FEA) of these sampling points in computer system to obtain numerically-simulated structural responses for each sampling point (i.e., each alternative design has a unique set of N design variables).        c. Use the numerically-simulated structural responses to construct approximations known as metamodels that can be used for predicting structural responses at any location within the N-dimensional design variable space.        d. Obtain a series of Pareto Optimal Points (i.e., product designs) or product designs X*={x1, x2, x3, . . . , xQ} by solving an approximate design optimization problem by minimizing the objective functions {tilde over (f)}i; i=1, 2, 3, . . . , n subject to the constraints {tilde over (g)}i(x)≦0; j=1, 2, 3 . . . , m, where {tilde over (f)}i(x) and {tilde over (g)}i(x) are based on the approximate functions or metamodels.        
The purpose of using metamodels is three-fold: (1) The first is to reduce the number of simulations required to conduct the optimization as compared to using a direct multi-objective optimization method such as well known NSGA-II (Non-dominated Sorting Genetic Algorithm-II). Direct optimization algorithms typically use thousands of simulations to converge whereas a metamodel-based scheme may only require hundreds; (2) The second motivation for metamodel optimization is that the metamodel can be further adjusted after optimization. For instance, changes can be made to the design formulation followed by a rapid re-optimization as long as all the responses used to assemble the new formulation are available; and (3) A third reason is that reliability-based design optimization can only be conducted using metamodels since other methods (e.g., Monte Carlo simulations) are infeasible due to the requirement of multiple direct simulations.
The above approximate design optimization procedure can be modified by means of a Sequential Optimization Procedure for Multi-objective optimization. This is done by adding sample design points iteratively and constructing new metamodels based on the existing numerically-simulated structural responses (from previous iterations) as well as currently obtained numerically-simulated structural responses at the new sampling points.
One prior art approach for single objective optimization is therefore to construct a sequential method in which points are added in each iteration, progressively closer to the optimum designs. This improves the accuracy in the neighborhood of the solution while expending less effort in regions which are far away from the solution.
For Multi-objective optimization, one exemplary prior art approach is summarized in the following steps:                1. Select sampling points in the N-dimensional design variable space by spacing them as far as possible from each other and from previously selected points (in the first iteration there would be no previously simulated points, but in further iterations there would be an increasing number of previously simulated points).        2. Conduct computer or numerical simulations at selected sampling points.        3. Use the numerically-simulated structural responses to construct the metamodels.        4. Obtain approximate POPs as set X* by solving an approximate optimization problem constructed from these metamodels.        5. Select new sampling points using the neighborhood of the POPs as basis and use the new and existing sampling points to repeat the steps 2-4.        
The problems for the above procedure exist, for example, at step 5, selecting new sampling points is conducted in the full N-dimensional design variable space. First, ad hoc procedures depended upon user experiences or knowledge are often required. Second, a number of unnecessary sampling points may be selected. When optimizing a design of an automobile based on crashworthiness, each computer simulation of a full car model (i.e., one sampling point having a unique combination of N design variables) requires tens of hours of a multi-processor computer system to perform. As a result, the above described procedure is too time and resource consuming thus impractical and infeasible, sometimes.
Furthermore, selected sampling points at each new iteration may not be diversified enough. As a result, the search for the Pareto Optimal Points may be conducted in the wrong location due to inaccurate approximations of the design criteria (i.e., sampling points not diversified enough). Diversification of the selected sampling points allows a wider early search with a gradually tightening Pareto Optimal Region to enhance convergence of the search. It would, therefore, be desirable to have a more effective and efficient procedure for selecting sampling points in a sequential multi-objective engineering design optimization of a product.